def compute_critical_values(P, Q, p, q, mdist, P_dist, Q_dist):
"""
Usage
-----
Compute all the critical values between trajectories P and Q
Parameters
----------
param P : px2 numpy_array, Trajectory P
param Q : qx2 numpy_array, Trajectory Q
param p : int, number of points in Trajectory P
param q : int, number of points in Trajectory Q
mdist : p x q numpy array, pairwise distance between points of trajectories t1 and t2
param P_dist: p x 1 numpy_array, distances between consecutive points in P
param Q_dist: q x 1 numpy_array, distances between consecutive points in Q
Returns
-------
cc : list, all critical values between trajectories P and Q
"""
origin = eucl_dist(P[0], Q[0])
end = eucl_dist(P[-1], Q[-1])
end_point = max(origin, end)
cc = set([end_point])
for i in range(p - 1):
for j in range(q - 1):
Lij = point_to_seg(Q[j], P[i], P[i + 1], mdist[i, j],
mdist[i + 1, j], P_dist[i])
if Lij > end_point:
cc.add(Lij)
Bij = point_to_seg(P[i], Q[j], Q[j + 1], mdist[i, j],
mdist[i, j + 1], Q_dist[j])
if Bij > end_point:
cc.add(Bij)
return sorted(list(cc))
def compute_critical_values(P, Q, p, q):
"""
Usage
-----
Compute all the critical values between trajectories P and Q
Parameters
----------
param P : px2 numpy_array, Trajectory P
param Q : qx2 numpy_array, Trajectory Q
param p : float, number of points in Trajectory P
param q : float, number of points in Trajectory Q
Returns
-------
cc : list, all critical values between trajectories P and Q
"""
origin = eucl_dist(P[0], Q[0])
end = eucl_dist(P[-1], Q[-1])
end_point = max(origin, end)
cc = set([end_point])
for i in range(p - 1):
for j in range(q - 1):
Lij = point_to_seg(Q[j], P[i], P[i + 1])
if Lij > end_point:
cc.add(Lij)
Bij = point_to_seg(P[i], Q[j], Q[j + 1])
if Bij > end_point:
cc.add(Bij)
return sorted(list(cc))
def compute_critical_values(P, Q, p, q):
"""
Usage
-----
Compute all the critical values between trajectories P and Q
Parameters
----------
param P : px2 numpy_array, Trajectory P
param Q : qx2 numpy_array, Trajectory Q
param p : float, number of points in Trajectory P
param q : float, number of points in Trajectory Q
Returns
-------
cc : list, all critical values between trajectories P and Q
"""
origin = eucl_dist(P[0], Q[0])
end = eucl_dist(P[-1], Q[-1])
end_point = max(origin, end)
cc = set([end_point])
for i in range(p - 1):
for j in range(q - 1):
Lij = point_to_seg(Q[j], P[i], P[i + 1])
if Lij > end_point:
cc.add(Lij)
Bij = point_to_seg(P[i], Q[j], Q[j + 1])
if Bij > end_point:
cc.add(Bij)
return sorted(list(cc))
def frechet(P, Q):
"""
Usage
-----
Compute the frechet distance between trajectories P and Q
Parameters
----------
param P : px2 numpy_array, Trajectory P
param Q : qx2 numpy_array, Trajectory Q
Returns
-------
frech : float, the frechet distance between trajectories P and Q
"""
p = len(P)
q = len(Q)
mdist = eucl_dist_traj(P, Q)
P_dist = map(lambda ip: eucl_dist(P[ip], P[ip + 1]), range(p - 1))
Q_dist = map(lambda iq: eucl_dist(Q[iq], Q[iq + 1]), range(q - 1))
cc = compute_critical_values(P, Q, p, q, mdist, P_dist, Q_dist)
eps = cc[0]
while (len(cc) != 1):
m_i = len(cc) / 2 - 1
eps = cc[m_i]
rep = decision_problem(P, Q, p, q, eps, mdist, P_dist, Q_dist)
if rep:
cc = cc[:m_i + 1]
else:
cc = cc[m_i + 1:]
frech = eps
return frech
def e_sspd(t1, t2):
"""
Usage
-----
The sspd-distance between trajectories t1 and t2.
The sspd-distance isjthe mean of the spd-distance between of t1 from t2 and the spd-distance of t2 from t1.
Parameters
----------
param t1 : len(t1)x2 numpy_array
param t2 : len(t2)x2 numpy_array
Returns
-------
sspd : float
sspd-distance of trajectory t2 from trajectory t1
"""
mdist = eucl_dist_traj(t1, t2)
l_t1 = len(t1)
l_t2 = len(t2)
t1_dist = map(lambda it1: eucl_dist(t1[it1], t1[it1 + 1]), range(l_t1 - 1))
t2_dist = map(lambda it2: eucl_dist(t2[it2], t2[it2 + 1]), range(l_t2 - 1))
sspd = (e_spd(t1, t2, mdist, l_t1, l_t2, t2_dist) + e_spd(t2, t1, mdist.T, l_t2, l_t1, t1_dist)) / 2
return sspd
def e_erp(t0,t1,g):
"""
Usage
-----
The Edit distance with Real Penalty between trajectory t0 and t1.
Parameters
----------
param t0 : len(t0)x2 numpy_array
param t1 : len(t1)x2 numpy_array
Returns
-------
dtw : float
The Dynamic-Time Warping distance between trajectory t0 and t1
"""
n0 = len(t0)
n1 = len(t1)
C=np.zeros((n0+1,n1+1))
C[1:,0]=sum(map(lambda x : abs(eucl_dist(g,x)),t0))
C[0,1:]=sum(map(lambda y : abs(eucl_dist(g,y)),t1))
for i in np.arange(n0)+1:
for j in np.arange(n1)+1:
derp0 = C[i-1,j] + eucl_dist(t0[i-1],g)
derp1 = C[i,j-1] + eucl_dist(g,t1[j-1])
derp01 = C[i-1,j-1] + eucl_dist(t0[i-1],t1[j-1])
C[i,j] = min(derp0,derp1,derp01)
erp = C[n0,n1]
return erp
def e_erp(t0, t1, g):
"""
Usage
-----
The Edit distance with Real Penalty between trajectory t0 and t1.
Parameters
----------
param t0 : len(t0)x2 numpy_array
param t1 : len(t1)x2 numpy_array
Returns
-------
dtw : float
The Dynamic-Time Warping distance between trajectory t0 and t1
"""
n0 = len(t0)
n1 = len(t1)
C = np.zeros((n0 + 1, n1 + 1))
gt0_dist = map(lambda x: abs(eucl_dist(g, x)), t0)
gt1_dist = map(lambda x: abs(eucl_dist(g, x)), t1)
mdist = eucl_dist_traj(t0, t1)
C[1:, 0] = sum(gt0_dist)
C[0, 1:] = sum(gt1_dist)
for i in np.arange(n0) + 1:
for j in np.arange(n1) + 1:
derp0 = C[i - 1, j] + gt0_dist[i - 1]
derp1 = C[i, j - 1] + gt1_dist[j - 1]
derp01 = C[i - 1, j - 1] + mdist[i - 1, j - 1]
C[i, j] = min(derp0, derp1, derp01)
erp = C[n0, n1]
return erp
def _c(ca,i,j,P,Q):
if ca[i,j] > -1:
return ca[i,j]
elif i == 0 and j == 0:
ca[i,j] = eucl_dist(P[0],Q[0])
elif i > 0 and j == 0:
ca[i,j] = max(_c(ca,i-1,0,P,Q),eucl_dist(P[i],Q[0]))
elif i == 0 and j > 0:
ca[i,j] = max(_c(ca,0,j-1,P,Q),eucl_dist(P[0],Q[j]))
elif i > 0 and j > 0:
ca[i,j] = max(min(_c(ca,i-1,j,P,Q),_c(ca,i-1,j-1,P,Q),_c(ca,i,j-1,P,Q)),eucl_dist(P[i],Q[j]))
else:
ca[i,j] = float("inf")
return ca[i,j]
def e_dtw(t0,t1):
"""
Usage
-----
The Dynamic-Time Warping distance between trajectory t0 and t1.
Parameters
----------
param t0 : len(t0)x2 numpy_array
param t1 : len(t1)x2 numpy_array
Returns
-------
dtw : float
The Dynamic-Time Warping distance between trajectory t0 and t1
"""
n0 = len(t0)
n1 = len(t1)
C=np.zeros((n0+1,n1+1))
C[1:,0]=float('inf')
C[0,1:]=float('inf')
for i in np.arange(n0)+1:
for j in np.arange(n1)+1:
C[i,j]=eucl_dist(t0[i-1],t1[j-1]) + min(C[i,j-1],C[i-1,j-1],C[i-1,j])
dtw = C[n0,n1]
return dtw
def discret_frechet(t0, t1):
"""
Usage
-----
Compute the discret frechet distance between trajectories P and Q
Parameters
----------
param t0 : px2 numpy_array, Trajectory t0
param t1 : qx2 numpy_array, Trajectory t1
Returns
-------
frech : float, the discret frechet distance between trajectories t0 and t1
"""
n0 = len(t0)
n1 = len(t1)
C = np.zeros((n0 + 1, n1 + 1))
C[1:, 0] = float('inf')
C[0, 1:] = float('inf')
for i in np.arange(n0) + 1:
for j in np.arange(n1) + 1:
C[i, j] = max(eucl_dist(t0[i - 1], t1[j - 1]), min(C[i, j - 1], C[i - 1, j - 1], C[i - 1, j]))
dtw = C[n0, n1]
return dtw
def e_lcss(t0, t1,eps):
"""
Usage
-----
The Longuest-Common-Subsequence distance between trajectory t0 and t1.
Parameters
----------
param t0 : len(t0)x2 numpy_array
param t1 : len(t1)x2 numpy_array
eps : float
Returns
-------
lcss : float
The Longuest-Common-Subsequence distance between trajectory t0 and t1
"""
n0 = len(t0)
n1 = len(t1)
# An (m+1) times (n+1) matrix
C = [[0] * (n1+1) for _ in range(n0+1)]
for i in range(1, n0+1):
for j in range(1, n1+1):
if eucl_dist(t0[i-1],t1[j-1])<eps:
C[i][j] = C[i-1][j-1] + 1
else:
C[i][j] = max(C[i][j-1], C[i-1][j])
lcss = 1-float(C[n0][n1])/min([n0,n1])
return lcss
def _c(ca, i, j, P, Q):
if ca[i, j] > -1:
return ca[i, j]
elif i == 0 and j == 0:
ca[i, j] = eucl_dist(P[0], Q[0])
elif i > 0 and j == 0:
ca[i, j] = max(_c(ca, i - 1, 0, P, Q), eucl_dist(P[i], Q[0]))
elif i == 0 and j > 0:
ca[i, j] = max(_c(ca, 0, j - 1, P, Q), eucl_dist(P[0], Q[j]))
elif i > 0 and j > 0:
ca[i, j] = max(
min(_c(ca, i - 1, j, P, Q), _c(ca, i - 1, j - 1, P, Q),
_c(ca, i, j - 1, P, Q)), eucl_dist(P[i], Q[j]))
else:
ca[i, j] = float("inf")
return ca[i, j]
def e_dtw(t0, t1):
"""
Usage
-----
The Dynamic-Time Warping distance between trajectory t0 and t1.
Parameters
----------
param t0 : len(t0)x2 numpy_array
param t1 : len(t1)x2 numpy_array
Returns
-------
dtw : float
The Dynamic-Time Warping distance between trajectory t0 and t1
"""
n0 = len(t0)
n1 = len(t1)
C = np.zeros((n0 + 1, n1 + 1))
C[1:, 0] = float('inf')
C[0, 1:] = float('inf')
for i in np.arange(n0) + 1:
for j in np.arange(n1) + 1:
C[i, j] = eucl_dist(t0[i - 1], t1[j - 1]) + min(
C[i, j - 1], C[i - 1, j - 1], C[i - 1, j])
dtw = C[n0, n1]
return dtw
def e_lcss(t0, t1, eps):
"""
Usage
-----
The Longuest-Common-Subsequence distance between trajectory t0 and t1.
Parameters
----------
param t0 : len(t0)x2 numpy_array
param t1 : len(t1)x2 numpy_array
eps : float
Returns
-------
lcss : float
The Longuest-Common-Subsequence distance between trajectory t0 and t1
"""
n0 = len(t0)
n1 = len(t1)
# An (m+1) times (n+1) matrix
C = [[0] * (n1 + 1) for _ in range(n0 + 1)]
for i in range(1, n0 + 1):
for j in range(1, n1 + 1):
if eucl_dist(t0[i - 1], t1[j - 1]) < eps:
C[i][j] = C[i - 1][j - 1] + 1
else:
C[i][j] = max(C[i][j - 1], C[i - 1][j])
lcss = 1 - float(C[n0][n1]) / min([n0, n1])
return lcss
def e_edr(t0, t1, eps):
"""
Usage
-----
The Edit Distance on Real sequence between trajectory t0 and t1.
Parameters
----------
param t0 : len(t0)x2 numpy_array
param t1 : len(t1)x2 numpy_array
eps : float
Returns
-------
edr : float
The Longuest-Common-Subsequence distance between trajectory t0 and t1
"""
n0 = len(t0)
n1 = len(t1)
# An (m+1) times (n+1) matrix
C = [[0] * (n1 + 1) for _ in range(n0 + 1)]
for i in range(1, n0 + 1):
for j in range(1, n1 + 1):
if eucl_dist(t0[i - 1], t1[j - 1]) < eps:
subcost = 0
else:
subcost = 1
C[i][j] = min(C[i][j - 1] + 1, C[i - 1][j] + 1, C[i - 1][j - 1] + subcost)
edr = float(C[n0][n1]) / max([n0, n1])
return edr
def e_edr(t0, t1,eps):
"""
Usage
-----
The Edit Distance on Real sequence between trajectory t0 and t1.
Parameters
----------
param t0 : len(t0)x2 numpy_array
param t1 : len(t1)x2 numpy_array
eps : float
Returns
-------
edr : float
The Longuest-Common-Subsequence distance between trajectory t0 and t1
"""
n0 = len(t0)
n1 = len(t1)
# An (m+1) times (n+1) matrix
C = [[0] * (n1+1) for _ in range(n0+1)]
for i in range(1, n0+1):
for j in range(1, n1+1):
if eucl_dist(t0[i-1],t1[j-1])<eps:
subcost = 0
else:
subcost = 1
C[i][j] = min(C[i][j-1]+1, C[i-1][j]+1,C[i-1][j-1]+subcost)
edr = float(C[n0][n1])/max([n0,n1])
return edr
def e_hausdorff(t1, t2):
"""
Usage
-----
hausdorff distance between trajectories t1 and t2.
Parameters
----------
param t1 : len(t1)x2 numpy_array
param t2 : len(t2)x2 numpy_array
Returns
-------
h : float, hausdorff from trajectories t1 and t2
"""
mdist = eucl_dist_traj(t1, t2)
l_t1 = len(t1)
l_t2 = len(t2)
t1_dist = map(lambda it1: eucl_dist(t1[it1], t1[it1 + 1]), range(l_t1 - 1))
t2_dist = map(lambda it2: eucl_dist(t2[it2], t2[it2 + 1]), range(l_t2 - 1))
h = max(e_directed_hausdorff(t1, t2, mdist, l_t1, l_t2, t2_dist),
e_directed_hausdorff(t2, t1, mdist.T, l_t2, l_t1, t1_dist))
return h
def ordered_mixed_distance(si,ei,sj,ej,siei,sjej,siei_norm_2,sjej_norm_2):
siei_norm=math.sqrt(siei_norm_2)
sjej_norm=math.sqrt(sjej_norm_2)
sisj=sj-si
siej=ej-si
u1=(sisj[0]*siei[0]+sisj[1]*siei[1])/siei_norm_2
u2=(siej[0]*siei[0]+siej[1]*siei[1])/siei_norm_2
ps=si+u1*siei
pe=si+u2*siei
cos_theta = max(-1,min(1,(sjej[0]*siei[0]+sjej[1]*siei[1])/(siei_norm*sjej_norm)))
theta = math.acos(cos_theta)
#perpendicular distance
lpe1=eucl_dist(sj,ps)
lpe2=eucl_dist(ej,pe)
if lpe1==0 and lpe2==0:
dped= 0
else:
dped = (lpe1*lpe1+lpe2*lpe2)/(lpe1+lpe2)
#parallel_distance
lpa1=min(eucl_dist(si,ps),eucl_dist(ei,ps))
lpa2=min(eucl_dist(si,pe),eucl_dist(ei,pe))
dpad=min(lpa1,lpa2)
#angle_distance
if 0 <= theta <HPI :
dad=sjej_norm * math.sin(theta)
elif HPI <= theta <= PI:
dad=sjej_norm
else:
raise ValueError("WRONG THETA")
fdist = (dped+dpad+dad)/3
return fdist
def ordered_mixed_distance(si, ei, sj, ej, siei, sjej, siei_norm_2, sjej_norm_2):
siei_norm = math.sqrt(siei_norm_2)
sjej_norm = math.sqrt(sjej_norm_2)
sisj = sj - si
siej = ej - si
u1 = (sisj[0] * siei[0] + sisj[1] * siei[1]) / siei_norm_2
u2 = (siej[0] * siei[0] + siej[1] * siei[1]) / siei_norm_2
ps = si + u1 * siei
pe = si + u2 * siei
cos_theta = max(-1, min(1, (sjej[0] * siei[0] + sjej[1] * siei[1]) / (siei_norm * sjej_norm)))
theta = math.acos(cos_theta)
# perpendicular distance
lpe1 = eucl_dist(sj, ps)
lpe2 = eucl_dist(ej, pe)
if lpe1 == 0 and lpe2 == 0:
dped = 0
else:
dped = (lpe1 * lpe1 + lpe2 * lpe2) / (lpe1 + lpe2)
# parallel_distance
lpa1 = min(eucl_dist(si, ps), eucl_dist(ei, ps))
lpa2 = min(eucl_dist(si, pe), eucl_dist(ei, pe))
dpad = min(lpa1, lpa2)
# angle_distance
if 0 <= theta < HPI:
dad = sjej_norm * math.sin(theta)
elif HPI <= theta <= PI:
dad = sjej_norm
else:
raise ValueError("WRONG THETA")
fdist = (dped + dpad + dad) / 3
return fdist
def free_line(p, eps, s):
"""
Usage
-----
Return the free space in the segment s, from point p.
This free space is the set of all point in s whose distance from p is at most eps.
Since s is a segment, the free space is also a segment.
We return a 1x2 array whit the fraction of the segment s which are in the free space.
If no part of s are in the free space, return [-1,-1]
Parameters
----------
param p : 1x2 numpy_array, centre of the circle
param eps : float, radius of the circle
param s : 2x2 numpy_array, line
Returns
-------
lf : 1x2 numpy_array
fraction of segment which is in the free space (i.e [0.3,0.7], [0.45,1], ...)
If no part of s are in the free space, return [-1,-1]
"""
px = p[0]
py = p[1]
s1x = s[0, 0]
s1y = s[0, 1]
s2x = s[1, 0]
s2y = s[1, 1]
if s1x == s2x and s1y==s2y:
if eucl_dist(p, s[0]) > eps:
lf = [-1, -1]
else:
lf = [0, 1]
else:
if point_to_seg(p, s[0], s[1]) > eps:
#print("No Intersection")
lf = [-1, -1]
else:
segl = eucl_dist(s[0], s[1])
segl2 = segl * segl
intersect = circle_line_intersection(px, py, s1x, s1y, s2x, s2y, eps)
if intersect[0][0] != intersect[1][0] or intersect[0][1] != intersect[1][1]:
i1x = intersect[0, 0]
i1y = intersect[0, 1]
u1 = (((i1x - s1x) * (s2x - s1x)) + ((i1y - s1y) * (s2y - s1y))) / segl2
i2x = intersect[1, 0]
i2y = intersect[1, 1]
u2 = (((i2x - s1x) * (s2x - s1x)) + ((i2y - s1y) * (s2y - s1y))) / segl2
ordered_point = sorted((0, 1, u1, u2))
lf = ordered_point[1:3]
else:
if px == s1x and py==s1y:
lf = [0, 0]
elif px == s2x and py==s2y:
lf = [1, 1]
else:
i1x = intersect[0][0]
i1y = intersect[0][1]
u1 = (((i1x - s1x) * (s2x - s1x)) + ((i1y - s1y) * (s2y - s1y))) / segl2
if 0 <= u1 <= 1:
lf = [u1, u1]
else:
lf = [-1, -1]
return lf
def free_line(p, eps, s, dps1, dps2, ds):
"""
Usage
-----
Return the free space in the segment s, from point p.
This free space is the set of all point in s whose distance from p is at most eps.
Since s is a segment, the free space is also a segment.
We return a 1x2 array whit the fraction of the segment s which are in the free space.
If no part of s are in the free space, return [-1,-1]
Parameters
----------
param p : 1x2 numpy_array, centre of the circle
param eps : float, radius of the circle
param s : 2x2 numpy_array, line
Returns
-------
lf : 1x2 numpy_array
fraction of segment which is in the free space (i.e [0.3,0.7], [0.45,1], ...)
If no part of s are in the free space, return [-1,-1]
"""
px = p[0]
py = p[1]
s1x = s[0, 0]
s1y = s[0, 1]
s2x = s[1, 0]
s2y = s[1, 1]
if s1x == s2x and s1y == s2y:
if eucl_dist(p, s[0]) > eps:
lf = [-1, -1]
else:
lf = [0, 1]
else:
if point_to_seg(p, s[0], s[1], dps1, dps2, ds) > eps:
# print("No Intersection")
lf = [-1, -1]
else:
segl = eucl_dist(s[0], s[1])
segl2 = segl * segl
intersect = circle_line_intersection(px, py, s1x, s1y, s2x, s2y,
eps)
if intersect[0][0] != intersect[1][0] or intersect[0][
1] != intersect[1][1]:
i1x = intersect[0, 0]
i1y = intersect[0, 1]
u1 = (((i1x - s1x) * (s2x - s1x)) + ((i1y - s1y) *
(s2y - s1y))) / segl2
i2x = intersect[1, 0]
i2y = intersect[1, 1]
u2 = (((i2x - s1x) * (s2x - s1x)) + ((i2y - s1y) *
(s2y - s1y))) / segl2
ordered_point = sorted((0, 1, u1, u2))
lf = ordered_point[1:3]
else:
if px == s1x and py == s1y:
lf = [0, 0]
elif px == s2x and py == s2y:
lf = [1, 1]
else:
i1x = intersect[0][0]
i1y = intersect[0][1]
u1 = (((i1x - s1x) * (s2x - s1x)) + ((i1y - s1y) *
(s2y - s1y))) / segl2
if 0 <= u1 <= 1:
lf = [u1, u1]
else:
lf = [-1, -1]
return lf